[Paper Review] Gerbe-holonomy for surfaces with defect networks

Ingo Runkel, Rafał R. Suszek|arXiv (Cornell University)|Aug 11, 2008

Black Holes and Theoretical Physics35 references4 citations

TL;DR

This paper introduces a holonomy formalism for two-dimensional sigma models with defect networks using gerbes with connection, deriving defect gluing conditions that distinguish conformal from topological defects. For the WZW model, both classical and quantum computations of holonomy yield cohomologous 3-cocycles on the center of the target group G, establishing a deep link between classical geometry and quantum field theory in defect networks.

ABSTRACT

We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW model with defects labelled by elements of the centre Z(G) of the target Lie group G; comparing the holonomy for different defect networks gives rise to a 3-cocycle on Z(G). Next, we describe the factorisation properties of two-dimensional quantum field theories in the presence of defects and compare the correlators for different defect networks in the quantum WZW model. This, again, results in a 3-cocycle on Z(G). We observe that the cocycles obtained in the classical and in the quantum computation are cohomologous.

Motivation & Objective

• To formulate the sigma-model action with embedded defect networks in the presence of a three-form field strength H.
• To derive consistent gluing conditions for fields and their derivatives at defect junctions using invariance under homotopy moves.
• To distinguish conformal from topological defects via the derived gluing conditions.
• To compute the holonomy for the WZW model with defects labeled by elements of the center Z(G), yielding a 3-cocycle on Z(G).
• To compare classical and quantum holonomy computations and show that the resulting 3-cocycles are cohomologous.

Proposed method

• Formulate the sigma-model action with a topological term S_top[X] as the logarithm of a U(1)-valued holonomy associated with a gerbe with connection on the target space M.
• Use local data (B, A, g) on patches, overlaps, and triple overlaps to define the holonomy, incorporating transition functions f and gauge potentials P.
• Define defect networks via bi-branes (for circular defects) and inter-bi-branes (for junctions), generalizing D-branes and gerbe modules to junction structures.
• Derive defect gluing conditions via invariance analysis of the action under homotopy moves of the defect network, leading to consistency conditions on field values and connections.
• Construct a new network-field configuration via a vertex move and compute the holonomy difference, showing invariance under such moves.
• Compare classical and quantum holonomy computations in the WZW model, demonstrating that the resulting 3-cocycles on Z(G) are cohomologous.

Experimental results

Research questions

• RQ1 How can the sigma-model action be consistently extended to world-sheets with embedded defect networks, including junctions, in the presence of a three-form field strength?
• RQ2 What are the necessary gluing conditions for the embedding field X and its derivatives at defect junctions, and how do they distinguish conformal from topological defects?
• RQ3 What is the geometric and algebraic structure of holonomy for defect networks involving multiple junctions and circular defects?
• RQ4 How does the holonomy computation in the classical WZW model with Z(G)-labeled defects yield a 3-cocycle on Z(G)?
• RQ5 Is the 3-cocycle obtained from classical holonomy cohomologous to the one derived from quantum correlators in the WZW model?

Key findings

• The holonomy for world-sheets with defect networks is consistently defined using gerbe data (B, A, g) and extended to junctions via inter-bi-branes, generalizing boundary and bi-brane holonomy.
• The defect gluing conditions derived from action invariance under homotopy moves ensure consistency of field configurations across junctions and distinguish conformal from topological defects.
• For the WZW model with defects labeled by elements of Z(G), the classical holonomy computation yields a 3-cocycle on Z(G), arising from the structure of the gerbe and transition functions.
• In the quantum WZW model, correlators for different defect networks also yield a 3-cocycle on Z(G), computed via factorization and symmetry properties of the theory.
• The 3-cocycle obtained from the classical holonomy computation is cohomologous to the one from the quantum computation, establishing a deep consistency between classical geometry and quantum field theory.
• The invariance of the action under vertex moves, verified via holonomy difference computation, confirms the consistency of the defect network formalism.